發(fā)布時(shí)間：2018-03-06 03:11

  本文選題：級(jí)數(shù)　切入點(diǎn)：數(shù)列　出處：《浙江理工大學(xué)》2016年碩士論文　論文類型：學(xué)位論文

【摘要】：在分析學(xué)中,三角級(jí)數(shù)和Fourier級(jí)數(shù)系數(shù)的單調(diào)遞減條件及其推廣是人們研究的焦點(diǎn)之一.1916年,英國(guó)學(xué)者Chaundy和Jolliffe對(duì)三角級(jí)數(shù)一致收斂性在單調(diào)條件下建立了一個(gè)經(jīng)典定理.隨后,許多學(xué)者繼續(xù)了這個(gè)方面的工作.單調(diào)性由此推廣到各種擬單調(diào)和各種有界變差條件之下.2005年,樂瑞君和周頌平提出了兼容這兩個(gè)方向的分組有界變差(GBV)的概念.后來,在最一般的均值有界變差概念提出之后,周頌平等人于2014年又將此推廣到了實(shí)意義條件下并在經(jīng)典分析中建立了許多重要的應(yīng)用.本文在前人的基礎(chǔ)上,對(duì)柯西并項(xiàng)準(zhǔn)則、強(qiáng)逼近及其相關(guān)嵌入定理進(jìn)行進(jìn)一步推廣研究.在研究過程中,將原有經(jīng)典定理中的條件推廣至實(shí)意義條件下,對(duì)復(fù)雜變號(hào)的區(qū)間采用巧妙的分割方法,以此來證明兩個(gè)定理并說明定理的最終適用范圍.此外,順便給出了幾乎單調(diào)遞減數(shù)列與分組有界變差數(shù)列互不包含的關(guān)系.全文共分為五章：第一章為緒論,首先介紹國(guó)內(nèi)外研究現(xiàn)狀,接著對(duì)論文涉及的相關(guān)符號(hào)一一給出定義,并闡述各種數(shù)列的概念.最后,簡(jiǎn)單地介紹論文的結(jié)構(gòu)框架.第二章針對(duì)Otto Szasz的柯西收斂準(zhǔn)則進(jìn)行推廣.此前,樂瑞君和解烈軍已經(jīng)將該定理推廣至非負(fù)的分組有界變差(GBV)條件并證明分組有界變差的不可減弱性.本章將經(jīng)典柯西并項(xiàng)準(zhǔn)則推廣的條件減弱到實(shí)意義下的分組有界變差(GBV*)條件,采用特殊的分割方法建立數(shù)列及積分的相關(guān)定理.同時(shí),舉例應(yīng)用該定理.在第三章中,主要對(duì)Tikhonov在擬單調(diào)(QM)及剩余有界變差(RBV)條件下的強(qiáng)逼近及其相關(guān)嵌入定理進(jìn)行研究.2010年,王敏芝已經(jīng)給出單邊、非負(fù)的均值有界變差(MVBV)條件下的定理.本章在第二章分割的基礎(chǔ)上進(jìn)一步細(xì)分,對(duì)分割作出了本質(zhì)性的推廣.最終,我們給出了實(shí)意義下修正的均值有界變差的強(qiáng)逼近及其相關(guān)嵌入定理的證明.雖然幾乎單調(diào)遞減數(shù)列(AMS)與分組有界變差數(shù)列(GBVS)之間互不包含的關(guān)系是顯而易見的,但這需要一個(gè)具體的證明過程.第四章將通過構(gòu)造平凡與非平凡數(shù)列的反例來證明此關(guān)系.最后,我們對(duì)全文進(jìn)行總結(jié)與展望.
[Abstract]:In the field of analysis, the monotone decreasing condition and its generalization of the coefficients of trigonometric series and Fourier series are one of the focuses of research. In 1916, the British scholars Chaundy and Jolliffe established a classical theorem on the uniform convergence of trigonometric series under monotonic conditions. Many scholars have continued their work in this field. Monotonicity is thus extended to various quasi-monotone and bounded variation conditions. In 2005, Le Ruijun and Zhou Songping put forward the concept of bounded variation in groups compatible with these two directions. After the most general concept of bounded mean variation was put forward, Zhou Songping and others extended it to the real meaning condition in 2014 and established many important applications in classical analysis. The strong approximation and its related embedding theorems are further generalized and studied. In the course of the research, the conditions in the original classical theorems are extended to the real meaning conditions, and the subdivision method is used for the interval of complex sign variation. In addition, the relation between the sequence of almost monotone decreasing numbers and the sequence of bounded variable number of groups is given. The whole paper is divided into five chapters: the first chapter is the introduction. This paper first introduces the current research situation at home and abroad, then defines the relevant symbols involved in the paper, and expounds the concepts of various series of numbers. In chapter 2, the Cauchy convergence criterion of Otto Szasz is generalized. The theorem has been extended to the condition of nonnegative bounded variation of grouping (GBV) and proved the irabligibility of bounded variation of grouping. In this chapter, the condition of the extension of the classical Cauchy complex criterion is reduced to the grouping in the real sense. Bounded variation condition, A special partition method is used to establish the relevant theorems of sequence and integral. At the same time, an example is given to apply the theorem. In this paper, the strong approximation and its related embedding theorems of Tikhonov under the condition of quasi monotone QM) and residual bounded variation are studied. In 2010, Wang Minzhi presented one-sided approximation. Theorem under the condition of bounded variation of nonnegative mean value MVBV). In this chapter, the segmentation is further subdivided on the basis of the second chapter, and the essential generalization of the segmentation is made. In this paper, we give the proof of the strong approximation of the modified mean bounded variation and its related embedding theorem in the real sense. Although the relation between the almost monotone decreasing sequence (AMS) and the grouped bounded variable difference sequence (GBVS) is obvious, However, this requires a concrete proof process. Chapter 4th will prove this relationship by constructing counterexample of ordinary and nontrivial sequence. Finally, we summarize and look forward to the full text.
【學(xué)位授予單位】：浙江理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：O173

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